Growth and Decay Factor

Introduction to growth and decay factor:

                     Exponential growth is nothing but the exponential function occurs if the rate of growth is proportional to the functions current value in themathematical functional part. In discrete domain it contains some equalintervals where it can be called as the geometric growth or geometric decay factor. The geometric factor value forms the geometric progression. Now we are going to see about the exponential growth and the decay factor.

Problem for Growth and Decay Factor:

                  Find the exponential growth if it takes $1200 to double at 21/2 % compounded continuously

Sol:

                First we have to take the formula of growth factor,

                              A = Pert

              The rate value which can be taken as 0.105

                         2400 = 1200 e 0.105t

             We have to take natural log on both sides we get,

                             2 = e0.105t

                        ln 2 = ln e 0.105t

                       ln 2 = 0.105t (ln e)

                       ln 2 = 0.105t

           Use the calculator to solve the logarithm values,

                       0.693147 = 0.105t

                               t = 6.666

          Thus it takes 6.66 years to double the money.

More Examples for Growth and Decay Factor:

Ex 1:

               Find the decay constant k where time t = 4 days, N = 900 and No = 1000 and find N after 7 days?

Sol:

               Let us take the decay formula

                     N = No . e kt

                     ln (N /No) = k . t

                ln (900 / 1000) = k . 4

                          -0.105 = 4k

                            k = -0.105/4

                            k = -0.026 day -1

               Thus we can calculate the Value of N after 7 days,

                         N = 1000 e -0.026 × 7

                         N = -999.818.

Ex: 2

            Find the exponential growth factor if it takes $1500to double at 41/4 % compounded continuously

Sol:

           First we have to take the formula,

                      A = Pert

         The rate value which can be taken as 0.105

                  3000 = 1500 e 0.125t

       We have to take natural log on both sides we get,

                       2 = e0.125t

                    ln 2 = ln e 0.125t

                   ln 2 = 0.125t (ln e)

                   ln 2 = 0.125t

            Use the calculator to solve the logarithm values,

                  0.693147 = 0.125t

                                 t = 5.54

          Thus it takes 5.54 years to double the money.